Complete -partite subgraphs of dense -graphs

Extending a classical result of Erdős, we derive the following concise statement: ≥ 3 and ( ln n ) − 1 / ( r − 1 ) ≤ α ≤ r − 3 . Then every r -uniform graph on n vertices with at least α n r / r ! edges contains a complete r -partite subgraph with r − 1 classes of size ⌊ α ( ln n ) 1 / ( r − 1 ) ⌋ and one class of size ⌈ n 1 − α r − 2 ⌉ . in result is a similar, but stronger statement about directed hypergraphs.